Problem: Captain Luis has a ship, the H.M.S Crimson Lynx. The ship is two furlongs from the dread pirate Umaima and her merciless band of thieves. The Captain has probability $\dfrac{4}{5}$ of hitting the pirate ship, if his ship hasn't already been hit. If it has been hit, he will always miss. The pirate has probability $\dfrac{1}{3}$ of hitting the Captain's ship, if her ship hasn't already been hit. If it has been hit, she will always miss as well. If the pirate shoots first, what is the probability that the pirate misses the Captain's ship, but the Captain hits?
The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened In this case, event A is the pirate missing the Captain's ship and event B is the Captain hitting the pirate ship. The pirate fires first, so her ship can't be sunk before she fires his cannons. So, the probability of the pirate missing the Captain's ship is $\dfrac{2}{3}$ If the pirate missed the Captain's ship, the Captain has a normal chance to fire back. So, the probability of the Captain hitting the pirate ship given the pirate missing the Captain's ship is $\dfrac{4}{5}$ The probability that the pirate misses the Captain's ship, but the Captain hits is then the probability of the pirate missing the Captain's ship times the probability of the Captain hitting the pirate ship given the pirate missing the Captain's ship. This is $\dfrac{2}{3} \cdot \dfrac{4}{5} = \dfrac{8}{15}$